Let `g(x)=x^(3)-4x+6.` If `f'(x)=g(x) and f(1)=2,` then what is f(x) equal to?

Options

A. `x^(3)-4x+3`
B. `x^(3)-4x+6`
C. `x^(3)-4x+1`
D. `x^(3)-4x+5`

Explanation

To solve the problem step by step, we start with the given function:

**Step 1: Write down the given function and its derivative.**
We have:
\[ g(x) = x^3 - 4x + 6 \]
We know that:
\[ f'(x) = g(x) \]

**Step 2: Integrate \( f'(x) \) to find \( f(x) \).**
To find \( f(x) \), we need to integrate \( g(x) \):
\[ f(x) = \int g(x) \, dx = \int (x^3 - 4x + 6) \, dx \]

**Step 3: Perform the integration.**
Integrating term by term:
\[ f(x) = \frac{x^4}{4} - 2x^2 + 6x + C \]
where \( C \) is the constant of integration.

**Step 4: Use the initial condition to find \( C \).**
We are given that \( f(1) = 2 \). Substituting \( x = 1 \) into the equation:
\[ f(1) = \frac{1^4}{4} - 2(1^2) + 6(1) + C = 2 \]
This simplifies to:
\[ \frac{1}{4} - 2 + 6 + C = 2 \]
\[ \frac{1}{4} + 4 + C = 2 \]
\[ C = 2 - 4 - \frac{1}{4} \]
\[ C = -2 - \frac{1}{4} = -\frac{8}{4} - \frac{1}{4} = -\frac{9}{4} \]

**Step 5: Substitute \( C \) back into \( f(x) \).**
Now we substitute \( C \) back into the expression for \( f(x) \):
\[ f(x) = \frac{x^4}{4} - 2x^2 + 6x - \frac{9}{4} \]

**Step 6: Simplify \( f(x) \).**
To express \( f(x) \) in a cleaner form:
\[ f(x) = \frac{x^4}{4} - 2x^2 + 6x - \frac{9}{4} \]

Thus, the final answer is:
\[ f(x) = \frac{x^4}{4} - 2x^2 + 6x - \frac{9}{4} \]

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